Introductory Microeconomics (ES10001)

Transcription

1 Introductory Microeconomic (ES10001) Exercie 2: Suggeted Solution 1. Match each lettered concept with the appropriate numbered phrae: (a) Cro price elaticity of demand; (b) inelatic demand; (c) long run; (d) budget hare; (e) normal good; (f) neceity; (g) ubtitute; (h) unit elatic demand; (i) hort-run (j) income elaticity of demand; (k) elatic demand; (l) inferior good; (m) total pending on a good; (n) complement; (o) luxury good; (p) own price elaticity of demand. Solution: (1) The % change in q d divided by correponding % change in income; (2) expenditure on a good a a proportion of total pending; (3) a 1% change in price lead to a le than 1% change in q d ; (4)a good with a poitive income elaticity of demand; (5) a good with a negative income elaticity of demand; (6) a meaure of the reponivene of demand for a good to a change in the price of another good; (7) a good having an income elaticity of demand le than 1; (8) the % change in the quantity of a good demanded divided by the correponding % change in it price; (9) two good are decribed thu if a rie in the price of one i generally aociated with an increae in demand for the other; (10) a good having an income elaticity of demand greater than 1; (11) quantity demanded of a good multiplied by it price; (12) a 1% change in price lead to a more than 1% change in q d ; (13) expenditure i unchanged when price fall; (14) the period neceary for complete adjutment to a price change; (15) two good are decribed thu when an increae in the price of one i generally aociated with a fall in demand for the other; (16) the period during which conumer are till in the proce of adjuting to a price change. (1) j (5) l (9) g (13) h (2) d (6) a (10) o (14) c (3) b (7) f (11) m (15) n (4) e (8) p (12) k (16) i 2. The citizen of Ruritania are intereted in only three thing: potatoe, bread and jam. How would a potato blight affect the live of the local producer and conumer. Solution: Aume that potatoe and bread are ubtitute for one another, bread and jam are complement for one another, and potatoe and jam are unrelated. Start from upply contraction in potato market. Price of potatoe rie o demand for bread increae, price of bread increae o demand for jam fall. There will be econd-order effect, but we will ignore them for implicity. 3. Recall Quetion 4 from Exercie 1: (a) Calculate the revenue of upplier in thi market. How much are conumer pending on the good? (b) Calculate the elaticity of demand between p = 2 and p = 3, and between p = 9 and p = 10. 1

2 (c) Aume now that upply i perfectly inelatic at q = 9. What are the equilibrium price and quantity in the market now? (d) How much are buyer pending in equilibrium? (e) What happen to producer revenue if the upply curve hift to one that i perfectly inelatic at q = 6? (f) Why? Solution: Linear model: q d q ( p) = p ( p) = p (a) In equilibrium q ( d p ) = p = p = q ( p ), which implie p = 15, q = 6.5 and Revenue = p q = 97.5 = Expenditure (i.e. no taxe). (b) Arc Elaticity: Ε = q 1 q 0 Δq 100% q q = p 1 p 0 Δp 100% p p = Δq Δp q p = Δq Δp p q where: q = ( q 0 + q 1 ) 2, p = ( p 0 + p 1 ) 2 ] and q 0 = q( p 0 ) i the demand for the good at p = p 0. Thu, between p = 2 and p = 3, we have Ε = 0.5( ) = ; between p = 9 and p = 10, we have Ε = 0.5( ) = (c) If q = 9 then q d Ε = = ( p ) = p = 9 = q p (d) Total expenditure = p q = 10 9 = 90. uch that p = 10, q = 9 and (e) Shift in q to q = 6. Thu, q ( d p ) = p = 6 = q ( p ) implying p = 16, and q = 6. Thu, producer revenue = p q = 16 6 = 96 implying that revenue (and o expenditure) in the market ha increaed. The intuition for thi i that the decline in q raie the equilibrium price and reduce the equilibrium quantity in the market. Since demand at the original equilibrium price of p = 10 i inelatic (recall: E = 0.56 < 1), then a 1% increae in p at p = 10 lead to a le than 1% fall in demand. Thu revenue will rie. Indeed, it can be hown that price ha been raied too far. The elaticity of demand at p = 16 i Ε = = 1.33 > 1, which i elatic. At thi price, producer would do better by cutting price. i.e. demand i elatic at p = 16 and o a 1% cut in p at thi price 2

3 would lead to a more than 1% increae in demand uch that firm revenue will rie. The optimum (i.e. revenue maximiing) upply for producer i where E = 1 vi: Ε = 0.5 ˆpˆq = 1 ˆp ˆp = ˆp = ˆp ˆp = 14 q = ( 14 ) = 7 Thi price and quantity lay half way along the demand curve and o enure unit elaticity. i.e. Ε = 0.5( 14 7) = 1 and revenue are equal to = p q = 14 7 = 98, which repreent the maximum revenue available to producer in thi market. 4. The market for ipad may be repreented by the following linear model: q d q ( p) = 10 2 p ( p) = p (a) Calculate the invere demand and upply curve for thi market. What i the interpretation of thee curve? (b) Calculate the equilibrium price and quantity in the market. (c) The Government wihe to raie ome tax revenue from the market. Aume it impoe a per unit ale tax (i.e. a tax on eller per unit old) of 2. Calculate the new equilibrium price and quantity. (d) Aume the eller in the market lobby the Government to have the ale tax removed. The Government relent and intead impoe a per unit purchae tax (i.e. a tax on buyer per unit purchaed) of 2. Calculate the new equilibrium price and quantity in thi market. (e) Were the eller lobbying effort worthwhile? Explain. Solution: (a) Calculate the invere demand and upply curve for thi market. What i the interpretation of thee curve? q d q ( p) = 10 2 p p d ( q) = 5 0.5q ( p) = p p ( q) = q 3

4 Invere demand and upply curve may be interpreted a chedule of conumer and producer reervation price - i.e. maximum (minimum) price willing to pay (receive). (b) Calculate the equilibrium price and quantity in the market. q ( d p ) = 10 2 p = p = q p 4 p = 8 p = 2 q = 6 (c) ( q) = p q Thu: + t = q. ( q ) * = q * = 5 0.5q * = p d q * q * = 4 * = 3 d (d) ( q) = p d q Thu: t = 3 0.5q p ( q ) = q = 3 0.5q = ( q ) q = 4 = 1 Thu equilibrium quantity i unaffected by who bear the burden of the tax. Equilibrium price i different but notice that in each cae buyer and eller pay and receive the ame total price per unit. Under the ale tax the equilibrium price i 3 with the buyer paying 3 in total and the eller receiving 3 per unit from the buyer, but paying 2 per unit to the government, yielding a net revenue per unit of 1. Under the purchae tax, the equilibrium price i 1 per unit with the eller receiving $1 per unit and the buyer paying 1 per unit to the eller and 2 per unit to the government, implying a total expenditure of 3 per unit. It doe not matter upon whom we impoe the tax - the relative burden will be determined only by the nature of the demand and upply curve. Notice that in thi example the burden of the tax i hared equally. Thi i not alway the cae. 4

5 5. Xaquane and Yullare are obcure but talented eighteenth-century painter. The world tock of Xaquane i 100 and the world tock of Yullare i 70. The demand for each painter work depend on it own price and the price of the other painter work. If i the price of Xaquane and p y i the price of Yullare, the demand function for Xaquane i p y and the demand function for Yullare i p y. What i the equilibrium price of Yullare painting? A. $5 B. $11 C. $12 D. $7 E. None of the above. Solution: q x d p ( x, p y ) = 101 3p x + 2 p y = 100 = q x 3p x = 1+ 2 p y And: q y d (, p y ) = 72 + p y = 70 = q y p y = 2 + Thu: 3p x = 1+ 2( 2 + ) 3 = = 5 Thu: p y = 2 + p y = p y = 7 6. Conider the following information for market demand and upply. Aume that quantity demanded in time period t, q t d, depend on the price in that period,, wherea the quantity 5

6 upplied in time period t, q t, depend of the price in the previou period, 1, Market demand i given by: q t d = 20 8 Market upply i given by: q t = (a) Find the relationhip between and 1 (i.e. price in thi period and the previou period). (b) What can you ay about the relationhip between and 1? (c) What are the main limitation of the cobweb model? Solution: (a) Auming equilibrium condition hold, in each period all upply will be old (and it will equal the quantity demanded) o that: q t d = 20 8 = = q t Thi implie: = The above equation decribe the relationhip between and 1, and how the periodby-period adjutment in price. (b) The equation alo how u that when and 1 are initially different, price change become maller and maller over time. Equation (1) i a firt order difference equation that decribe price adjutment over time. When the coefficient of 1 lie between 0 and -1 (e.g ), the price adjutment proce converge to equilibrium overtime. When the coefficient on 1 i le than -1 (e.g ), the price adjutment proce diverge. (c) The main limitation of the cobweb model i it aumption of hortightedne on the part of producer. Conequently, it implie that producer are unable to evaluate market condition accurately and they do not learn from their mitake in term of pricing deciion from previou period, which reult in urplue and hortage. A a reult, the cobweb model i applicable when upply repond to price with a lag and when producer are hortighted. Further Comment: The tory that generally decribe the cobweb model i the following: Producer mut make their output deciion one period in advance of the actual ale for example, in agricultural production, planting precede by ome coniderable time the harveting and ale of output. Aume that the output deciion in period t i baed on the then-prevailing price,. Since thi output will not be available for ale until period t +1, however, then will determine 6

7 or, not q t but q t+1. Thu, we have a lagged upply function of the form q t+1 = q equivalently, by hifting the time ub-cript back by one period, q t = q ( 1 ). If conumer bae their demand deciion in each period on that period price level uch that q d t = q ( ), then intereting dynamic pattern will reult. Taking linear verion of thee (lagged) upply and (unlagged) demand function and auming that in each period the market price i alway et at a level which clear the market, then we have a market model with the following equation: q t d = a b q t d = q t q t = c + d 1 where a,b,d > 0 and a > c. 1 Thu, if the market clear in each period we mut have: q t d = a b = c + d 1 = q t which implie a ingle firt-order difference equation: = a c b = α β 1 d b 1 b > 0 and β = ( d b ) > 0. The model converge to equilibrium if < 1 d < b, diverge continuouly (i.e. explode) if β > 1 ( d b) > 1 d > b = 1 d = b. Thu, tability i enured if the where α = a c β < 1 d b and ocillate uniformly if β = 1 d b (normal) upply curve i tepper than the (normal) demand curve. Long-run equilibrium when β < 1 occur when the price doe not change from one period to the next vi = 1 = p. Thi implie: p = α β p = p = p = a c b a c b + d α 1+ β 1 = 1+ ( d b) a c b b b + d 1 The aumption that a > c i made to enure that long-run equilibrium (i.e. teady tate) price i poitive. It i ometime aumed further that c < 0 uch that the invere upply price i poitive for all q > 0 ee Figure 1 following. 7

8 See Figure 1 for an example of a convergent cobweb model: p a/b S p 0 p 2 p 4 p * p 3 p 1 c/d 0 a q q 2 q 4 q * q 3 q 1 D Figure 1: Convergent Cobweb Model Thu, in Figure 1 the pinning proce produce a centripetal cobweb. Auming an initial price, p 0 > p, we find from the upply curve that the quantity upplied in the following period (i.e. period 1) will be q 1. In order to clear the market, the quantity demanded in period 1 mut alo be q 1, which i poible if and only if the price i et at the level of p 1. Now, via the upply curve, the price p 1 will imply a upply of q 2 a the quantity upplied in period 2, which according to the demand curve will require a price of p 2 to be old uch that the market clear. Repeating thi reaoning, we can trace out the price and quantitie in ubequent period by imply following the arrowhead in the Figure, thereby pinning a cobweb around the demand and upply curve ever cloer to the long-run equilibrium price. It i apparent that whilt the price path i ocillatory, it i convergent. Table 1 following illutrate three example of a cobweb model where p 0 = 1, α = 2 and quantity p,q and β (i.e. the coefficient on 1 ) i given by 0.25, 1.25 and 1.00 repectively 8

9 Table 1: Cobweb Model = = = t Notice that where b = 0.25, long run (i.e. teady-tate) equilibrium 16 period when price i equal to 1.6. Thi can be confirmed from the previou teady tate equation and the coefficient from the original demand and upply function given in the quetion. = a c b + d = = In a two good world, if the own price elaticity demand of good x i greater than unity [i.e. E x = ( Δx )( x) > 1], then x and y mut be ubtitute for one another: TRUE / FALSE Solution: Conider the original and new budget contraint following a mall change in the price of good x: x + p y y = m x + p y y = m Thu: 9

10 x x + p y y p y y = 0 x x + x x = p y y + p y y which implie: = p y ( y y) x + x x x + Δx = p y Δy x + Δx p = Δy p x Δp y x Δx x Δx x = 1 Δy p y x = 1+ Δy p y x Ε x = 1+ Δy p y x > 1 which require Δy > 0 - i.e. an increae in the price of good x lead to an increae in the demand for good y uch that the two good are ubtitute. Intuitively, if Ε x > 1 then a rie in the price of good x (i.e. = > 0 ) will lead to a decreae in expenditure on good x. But ince nominal income i unchanged, reduced pending on good x implie increaed pending on good y if the budget contraint i to be atified which, a the price of good y i unchanged, implie Δy = y y > 0 - i.e. the demand for good y mut increae uch that the two good are ubtitute for one another. Alternatively, uing calculu and point elaticitie: Ε x = x x > 1, And: x + p y y = m Thu, holding y and m contant whilt conidering change in : 10

11 m = + p y x y = 0 1 x + x + p y y = 0 dr dp = dpq p dp = q p + dq ( p ) dp p = q p 1+ dq ( p ) dp p = q p q( p ) 1 Ε = 0, p 11